## Abstract

We consider coGapSVP_{√n,} a gap version of the shortest vector in a lattice problem. This problem is known to be in AM ∩ coNP but is not known to be in NP or in MA. We prove that it lies inside QMA, the quantum analogue of NP. This is the first non-trivial upper bound on the quantum complexity of a lattice problem. The proof relies on two novel ideas. First, we give a new characterization of QMA, called QMA+. Working with the QMA+ formulation allows us to circumvent a problem which arises commonly in the context of QMA: the prover might use entanglement between different copies of the same state in order to cheat. The second idea involves using estimations of autocorrelation functions for verification. We make the important observation that autocorrelation functions are positive definite functions and using properties of such functions we severely restrict the prover's possibility to cheat. We hope that these ideas will lead to further developments in the field.

Original language | English |
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Title of host publication | Proceedings - 44th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2003 |

Publisher | IEEE Computer Society |

Pages | 210-219 |

Number of pages | 10 |

ISBN (Electronic) | 0769520405 |

DOIs | |

State | Published - 2003 |

Event | 44th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2003 - Cambridge, United States Duration: 11 Oct 2003 → 14 Oct 2003 |

### Publication series

Name | Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS |
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Volume | 2003-January |

ISSN (Print) | 0272-5428 |

### Conference

Conference | 44th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2003 |
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Country/Territory | United States |

City | Cambridge |

Period | 11/10/03 → 14/10/03 |

### Bibliographical note

Publisher Copyright:© 2003 IEEE.

## Keywords

- Algorithm design and analysis
- Autocorrelation
- Computer science
- Cryptography
- Lattices
- Polynomials
- Quantum computing
- Quantum mechanics
- Upper bound